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N9

Calculate with and interpret standard form A × 10^n, where 1 ≤ A < 10 and n is an integer

Standard form

Worked answers, methods and verified real exam appearances for N9 on Edexcel GCSE Maths 1MA1.

Definition

Specification N9: calculate with and interpret standard form A×10nA\times10^n, where 1≤A<101\leq A<10 and nn is an integer, with and without a calculator.

Explanation

  • A positive index moves the decimal point right, so 5.3×104=53 0005.3\times10^4=53\,000; a negative index moves it left, so 7.4×10−5=0.000 0747.4\times10^{-5}=0.000\,074. To multiply, multiply the AA values and add the indices; to divide, divide the AA values and subtract the indices; then normalise, because 24×10−224\times10^{-2} is not standard form until it becomes 2.4×10−12.4\times10^{-1}.
  • To add or subtract, write both numbers with the same power of ten first, or convert to ordinary numbers: 9.7×106+2.45×107=0.97×107+2.45×107=3.42×1079.7\times10^6+2.45\times10^7=0.97\times10^7+2.45\times10^7=3.42\times10^7.
  • Adding the AA values when the powers differ is the standard error and scores nothing.
  • A method mark is available for a correct value in a non-standard form such as 34.2×10634.2\times10^6, and a final answer left as an ordinary number when standard form was asked for loses the last mark.
  • To order numbers, put them all in the same form first.

Worked example

1

Work out 3.6×10−4+7.5×10−51.5×103\dfrac{3.6\times10^{-4}+7.5\times10^{-5}}{1.5\times10^{3}}. Give your answer in standard form.

(3)

(Total for Question 1 is 3 marks)

Working

  1. 1.Write the numerator with a common power of ten: 3.6×10−4+0.75×10−4=4.35×10−43.6\times10^{-4}+0.75\times10^{-4}=4.35\times10^{-4}.
  2. 2.Divide the AA values and subtract the indices: 4.351.5×10−4−3=2.9×10−7\dfrac{4.35}{1.5}\times10^{-4-3}=2.9\times10^{-7}.
  3. 3.Check the coefficient: 1≤2.9<101\leq2.9<10, so the answer is in standard form.

Answer: 2.9×10−72.9\times10^{-7}

Mark scheme

Mark scheme for question 1
QuestionAnswerMarkAdditional guidance
1for 4.35×10−44.35\times10^{-4} or 0.0004350.000435 or both numbers in a form ready for addition, eg 3.6×10−4+0.75×10−43.6\times10^{-4}+0.75\times10^{-4}M1oe
for 4.35÷1.5×10−4−34.35\div1.5\times10^{-4-3} or 2.9×10n2.9\times10^n (n≠−7n\neq-7) or 0.000 000 290.000\,000\,29M1Correct answer in incorrect form
for 2.9×10−72.9\times10^{-7}A1cao

Common mistakes

  • Don't leave 24×10424\times10^4 as the final answer even though 2424 is not between 11 and 1010.
  • Don't add the indices when dividing powers of 1010.
  • Don't add coefficients whose powers of 1010 have not first been made equal.

Mark-scheme wording to use

  • “for 9 700 000 + 24 500 000 (= 34 200 000) or 3.42 × 10^n (n ≠ 7) oe or 3.4 × 10^7 or correct answer in incorrect form eg 34.2 × 10^6 or both in a form ready for addition, eg 9.7 × 10^6 + 24.5 × 10^6” — Pearson 1MA1, June 2024 Paper 3H, Q2(b)
  • “M1 for 4×6×10^(3–5) or 0.24 oe eg 24 × 10^–2 or 2.4×10^n where n ≠ –1 A1 cao” — Pearson 1MA1, June 2022 Paper 1H, Q4(c)
  • “for correct conversions to same format, condoning one error” — Pearson 1MA1, June 2020 Paper 1H, Q9

What the examiner wants

The powers dealt with before the coefficients. The June 2024 Paper 3H report says most reached 34 200 00034\,200\,000 on a calculator but many could not then write it in standard form, and that those scoring nothing had ignored the differing powers of ten, adding 9.79.7 to 2.452.45 to get 12.1512.15 and attaching a power of ten to that.

Exam tip

Circle the final coefficient and check 1≤A<101\leq A<10 before submitting any standard-form answer.

Worked practice

Q1
Tier 1 · Easy

1

Write 0.0000720.000072 in standard form.

(1)

(Total for Question 1 is 1 mark)

Mark scheme

Mark scheme for question 1
QuestionAnswerMarkMark scheme
1
  • 7.2×10−57.2\times10^{-5}
1Move the decimal point 55 places right to make 7.27.2. Moving right gives a negative power, so 0.000072=7.2×10−50.000072=7.2\times10^{-5}.
Q2
Tier 2 · Standard

2

Work out (6×107)(4×10−3)(6\times10^7)(4\times10^{-3}). Give your answer in standard form.

(2)

(Total for Question 2 is 2 marks)

Mark scheme

Mark scheme for question 2
QuestionAnswerMarkMark scheme
2
  • 2.4×1052.4\times10^5
2Multiply the factors and add the indices: 6×4=246\times4=24 and 107×10−3=10410^7\times10^{-3}=10^4. Thus the product is 24×104=2.4×10524\times10^4=2.4\times10^5.
Q3
Tier 3 · Hard

3

Work out 3.6×10−4+7.5×10−51.5×103\dfrac{3.6\times10^{-4}+7.5\times10^{-5}}{1.5\times10^3}. Give your answer in standard form.

(4)

(Total for Question 3 is 4 marks)

Mark scheme

Mark scheme for question 3
QuestionAnswerMarkMark scheme
3
  • 2.9×10−72.9\times10^{-7}
4Express the numerator with a common power: 3.6×10−4+0.75×10−4=4.35×10−43.6\times10^{-4}+0.75\times10^{-4}=4.35\times10^{-4}. Divide the factors and subtract the indices: (4.35/1.5)×10−4−3=2.9×10−7(4.35/1.5)\times10^{-4-3}=2.9\times10^{-7}.
Q4
Tier 1 · Easy

4

Write 4 530 0004\,530\,000 in standard form.

(1)

(Total for Question 4 is 1 mark)

Mark scheme

Mark scheme for question 4
QuestionAnswerMarkMark scheme
4
  • 4.53×1064.53\times10^6
1Move the decimal point 66 places left to make 4.534.53, so 4 530 000=4.53×1064\,530\,000=4.53\times10^6.
Q5
Tier 2 · Standard

5

Work out 8.4×1062.1×102\dfrac{8.4\times10^6}{2.1\times10^2}. Write your answer in the form A×10nA\times10^n, where 1≤A<101\leq A<10.

(2)

(Total for Question 5 is 2 marks)

Mark scheme

Mark scheme for question 5
QuestionAnswerMarkMark scheme
5
  • 4×1044\times10^4
2Divide the numbers and subtract the indices: (8.4÷2.1)×106−2=4×104(8.4\div2.1)\times10^{6-2}=4\times10^4.
Q6
Tier 3 · Hard

6

Machine A makes 4.5×1054.5\times10^5 components and machine B makes 0.62×1060.62\times10^6 components. Work out how many more components machine B makes. Give your answer in standard form.

(3)

(Total for Question 6 is 3 marks)

Mark scheme

Mark scheme for question 6
QuestionAnswerMarkMark scheme
6
  • 1.7×1051.7\times10^5 components
3Write both amounts with the same power: 0.62×106=6.2×1050.62\times10^6=6.2\times10^5. The difference is (6.2−4.5)×105=1.7×105(6.2-4.5)\times10^5=1.7\times10^5 components.
Q7
Tier 2 · Standard

7

A parcel has mass 3.75×1043.75\times10^4 grams. Another parcel has mass 86008600 grams. Work out their total mass in standard form.

(3)

(Total for Question 7 is 3 marks)

Mark scheme

Mark scheme for question 7
QuestionAnswerMarkMark scheme
7
  • 4.61×1044.61\times10^4 grams
38600=0.86×1048600=0.86\times10^4. The total is therefore (3.75+0.86)×104=4.61×104(3.75+0.86)\times10^4=4.61\times10^4 grams.
Q8
Tier 3 · Hard

8

Each tray contains 2.4×1032.4\times10^3 screws. There are 3.5×1023.5\times10^2 trays. After inspection, 6.7×1046.7\times10^4 screws are rejected. Work out the number of screws that are not rejected. Give your answer in standard form.

(4)

(Total for Question 8 is 4 marks)

Mark scheme

Mark scheme for question 8
QuestionAnswerMarkMark scheme
8
  • 7.73×1057.73\times10^5 screws
4The trays contain (2.4×103)(3.5×102)=8.4×105(2.4\times10^3)(3.5\times10^2)=8.4\times10^5 screws. Write the rejected amount as 0.67×1050.67\times10^5. The number left is (8.4−0.67)×105=7.73×105(8.4-0.67)\times10^5=7.73\times10^5 screws.
Q9
Tier 3 · Hard

9

A survey stores 4.2×1074.2\times10^7 readings. Each reading uses 1.5×10−31.5\times10^{-3} MB. After 1.1×1041.1\times10^4 MB is deleted, the remaining data is shared equally between 8×1028\times10^2 drives. Work out the amount on each drive. Give your answer in standard form.

(4)

(Total for Question 9 is 4 marks)

Mark scheme

Mark scheme for question 9
QuestionAnswerMarkMark scheme
9
  • 6.5×1016.5\times10^1 MB
4The readings use (4.2×107)(1.5×10−3)=6.3×104(4.2\times10^7)(1.5\times10^{-3})=6.3\times10^4 MB. After deletion, 6.3×104−1.1×104=5.2×1046.3\times10^4-1.1\times10^4=5.2\times10^4 MB remains. Each drive receives 5.2×1048×102=0.65×102=6.5×101\dfrac{5.2\times10^4}{8\times10^2}=0.65\times10^2=6.5\times10^1 MB.
Q10
Tier 3 · Hard

10

Factory X makes 3.2×1043.2\times10^4 clips each day for 1818 days. Factory Y makes 5.5×1055.5\times10^5 clips in total. Work out how many more clips factory X makes than factory Y. Give your answer in standard form.

(4)

(Total for Question 10 is 4 marks)

Mark scheme

Mark scheme for question 10
QuestionAnswerMarkMark scheme
10
  • 2.6×1042.6\times10^4 clips
4Factory X makes (3.2×104)×18=57.6×104=5.76×105(3.2\times10^4)\times18=57.6\times10^4=5.76\times10^5 clips. The difference is 5.76×105−5.5×105=0.26×105=2.6×1045.76\times10^5-5.5\times10^5=0.26\times10^5=2.6\times10^4 clips.

Verified exam appearances

SeriesPaperQuestionMarksCalculatorTierLinks
2019-062FQ272AllowedFoundationQPMS
2022-113HQ93AllowedHigherQPMS
2019-112HQ72AllowedHigherQPMS
2022-113FQ293AllowedFoundationQPMS
2019-062HQ82AllowedHigherQPMS
2024-063FQ234AllowedFoundationQPMS
2023-113FQ202AllowedFoundationQPMS
2023-063FQ284AllowedFoundationQPMS
2022-061HQ44Non-calculatorHigherQPMS
2024-113HQ22AllowedHigherQPMS
2019-113HQ73AllowedHigherQPMS
2024-113FQ182AllowedFoundationQPMS
2024-063HQ193AllowedHigherQPMS
2019-063HQ113AllowedHigherQPMS
2019-113FQ283AllowedFoundationQPMS
2021-113FQ234AllowedFoundationQPMS
2022-061FQ264Non-calculatorFoundationQPMS
2019-112FQ272AllowedFoundationQPMS
2024-112FQ272AllowedFoundationQPMS
2024-063HQ24AllowedHigherQPMS
2021-113HQ34AllowedHigherQPMS
2024-112HQ82AllowedHigherQPMS
2023-113HQ12AllowedHigherQPMS

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